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Adaptive Test Procedure for High Dimensional Regression Coefficient

Ping Zhao, Fengyi Song, Huifang Ma

TL;DR

The paper introduces an adaptive omnibus test for high-dimensional regression coefficients based on top-k L-statistics that interpolate between max-type and sum-type tests. By establishing joint weak convergence and asymptotic independence between extreme and standard components, it justifies combining multiple k values through a Cauchy transform and employing a wild bootstrap for calibration. The approach delivers accurate size control and strong power across sparse and dense alternatives, including non-Gaussian designs, and offers a practical, computationally simple framework. Overall, it provides a principled method to adapt to unknown sparsity in high-dimensional testing with theoretical guarantees and robust empirical performance.

Abstract

We develop a unified $L$-statistic testing framework for high-dimensional regression coefficients that adapts to unknown sparsity. The proposed statistics rank coordinate-wise evidence measures and aggregate the top $k$ signals, bridging classical max-type and sum-type tests. We establish joint weak convergence of the extreme-value component and standardized $L$-statistics under mild conditions, yielding an asymptotic independence that justifies combining multiple $k$'s. An adaptive omnibus test is constructed via a Cauchy combination over a dyadic grid of $k$, and a wild bootstrap calibration is provided with theoretical guarantees. Simulations demonstrate accurate size and strong power across sparse and dense alternatives, including non-Gaussian designs.

Adaptive Test Procedure for High Dimensional Regression Coefficient

TL;DR

The paper introduces an adaptive omnibus test for high-dimensional regression coefficients based on top-k L-statistics that interpolate between max-type and sum-type tests. By establishing joint weak convergence and asymptotic independence between extreme and standard components, it justifies combining multiple k values through a Cauchy transform and employing a wild bootstrap for calibration. The approach delivers accurate size control and strong power across sparse and dense alternatives, including non-Gaussian designs, and offers a practical, computationally simple framework. Overall, it provides a principled method to adapt to unknown sparsity in high-dimensional testing with theoretical guarantees and robust empirical performance.

Abstract

We develop a unified -statistic testing framework for high-dimensional regression coefficients that adapts to unknown sparsity. The proposed statistics rank coordinate-wise evidence measures and aggregate the top signals, bridging classical max-type and sum-type tests. We establish joint weak convergence of the extreme-value component and standardized -statistics under mild conditions, yielding an asymptotic independence that justifies combining multiple 's. An adaptive omnibus test is constructed via a Cauchy combination over a dyadic grid of , and a wild bootstrap calibration is provided with theoretical guarantees. Simulations demonstrate accurate size and strong power across sparse and dense alternatives, including non-Gaussian designs.
Paper Structure (11 sections, 3 theorems, 46 equations, 1 figure, 1 table)

This paper contains 11 sections, 3 theorems, 46 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Test Procedures
  3. Fixed parameter $k$
  4. Diverging parameter $k=\lceil \gamma m\rceil$
  5. Adaptive procedure
  6. Simulation
  7. Conclusion
  8. Appendix
  9. Proof of Theorem \ref{['th1']}
  10. Proof of Theorem \ref{['th2']}
  11. Proof of Theorem \ref{['th3']}

Key Result

Theorem 1

Suppose Assumptions (A1)-(A4) hold. Then as $\min(n,p)\to\infty$, we have (i) for all integer $1\le s\le m$ and $x\in\mathbb{R}$, (ii) for all integer $2\le k\le m$ and $x_1\ge \ldots\ge x_k\in\mathbb{R}$ where $b_m=2\log m-\log (\log m)$ and $\Lambda(x)=\exp\{-\pi^{-1/2}\exp (-x/2)\}$.

Figures (1)

  • Figure 1: Power curves of each test with different distributions under $n=100,p=200$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3